# Tagged: positive eigenvalue

## Problem 397

Suppose $A$ is a positive definite symmetric $n\times n$ matrix.

(a) Prove that $A$ is invertible.

(b) Prove that $A^{-1}$ is symmetric.

(c) Prove that $A^{-1}$ is positive-definite.

(MIT, Linear Algebra Exam Problem)

## Problem 396

A real symmetric $n \times n$ matrix $A$ is called positive definite if
$\mathbf{x}^{\trans}A\mathbf{x}>0$ for all nonzero vectors $\mathbf{x}$ in $\R^n$.

(a) Prove that the eigenvalues of a real symmetric positive-definite matrix $A$ are all positive.

(b) Prove that if eigenvalues of a real symmetric matrix $A$ are all positive, then $A$ is positive-definite.

## Problem 12

Let $A$ be an $n \times n$ real matrix. Prove the followings.

(a) The matrix $AA^{\trans}$ is a symmetric matrix.

(b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

(c) The matrix $AA^{\trans}$ is non-negative definite.

(An $n\times n$ matrix $B$ is called non-negative definite if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)

(d) All the eigenvalues of $AA^{\trans}$ is non-negative.